Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(9,10,13,11)(12,14,16,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,31)(30,32)(33,34,38,35)(36,39,44,41)(37,42,45,40)(43,48,46,47);(9,11,13,10)(12,14,16,15)(18,19)(20,27)(21,29)(31,32)(34,35)(37,45)(39,41)(43,46);(1,2,5,4)(3,6,7,8)(9,12)(10,14)(11,15)(13,16)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,31,30,32)(33,35,38,34)(36,39,44,41)(37,42,45,40)(43,47,46,48);(1,3,5,7)(2,6,4,8)(17,20,22,27)(18,23,19,25)(21,30,29,28)(24,32,26,31)(33,36)(34,39)(35,41)(37,46)(38,44)(40,48)(42,47)(43,45);(1,4,5,2)(3,8,7,6)(17,21,22,29)(18,24,19,26)(20,28,27,30)(23,31,25,32)(33,37)(34,40)(35,42)(36,43)(38,45)(39,47)(41,48)(44,46)Polycyclic presentation:
2^9|a^2=gh,b^2=fh,c^2=fi,d^2=gi,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3211023);